Non-overlapping Domain Decomposition Algorithm Research Articles

Non-overlapping domain decomposition algorithms with only primal velocity unknowns for the discontinuous viscosity Stokes problem

In this paper, non-overlapping domain decomposition algorithms for the Stokes problem with a discontinuous viscosity coefficient are proposed and analyzed. A pair of inf–sup stable finite element spaces with discontinuous pressure finite element functions is employed to obtain a discrete Stokes system. By introducing a non-overlapping subdomain partition for the given finite element spaces, the discrete Stokes system is reformulated into an algebraic system on dual unknowns, where the continuity on the velocity unknowns at the subdomain vertices is enforced strongly and the continuity on the remaining interface velocity unknowns is enforced weakly using the dual unknowns. The resulting algebraic system on the dual unknowns is then solved by an iterative method. The coarse problem in the resulting algebraic system is only related to the strongly coupled primal velocity unknowns and it is thus obtained as a symmetric positive definite matrix equation. To accelerate the iteration convergence, practical versions of lumped and Dirichlet preconditioners are proposed and their condition numbers are also analyzed for the resulting dual algebraic system. Numerical results are also included to confirm our condition number estimates.

Optimized Schwarz Methods for the Cahn–Hilliard Equation

The Cahn–Hilliard equation was originally proposed to describe the phase separation phenomenon for a binary alloy in the quenching process and now has been widely applied in many other scientific fields. To investigate its solution behavior, one mainly depends on the numerical simulations because of the nonlinearity. However, it would require very small time steps to describe the phase separation procedure since it evolves very quickly in time, which implies that an efficient solver for the spatial problem at each time level is very important. In this paper, we investigate a coupled second order elliptic system of constant coefficients, which occurs in many different unconditionally energy stable time discretization schemes for the Cahn–Hilliard equation. For a min-max problem of convergence factor that is obtained via Fourier analysis in the case of two-subdomain domain decomposition, by using an asymptotic analysis we obtain the optimized transmission parameters in explicit form applied in the Robin and the two-sided Robin transmission conditions for both the overlapping and the nonoverlapping Schwarz domain decomposition algorithms, and we obtain as well the corresponding asymptotic convergence rates. Particularly, we also explore and analyze the many-subdomain cases directly and find that the algorithms are scalable until a certain number of subdomains is reached for small time steps, even if there is no coarse grid correction. We finally use numerical examples to illustrate our theoretical findings.

Non-Overlapping Schwarz Waveform-Relaxation for Nonlinear Advection-Diffusion Equations

Nonlinear advection-diffusion equations often arise in the modeling of transport processes. We propose for these equations a non-overlapping domain decomposition algorithm of Schwarz waveform-relaxation (SWR) type. It relies on nonlinear zeroth-order (or Robin) transmission conditions between the sub-domains that ensure the continuity of the converged solution and of its normal flux across the interface. We prove existence of unique iterative solutions and the convergence of the algorithm. We then present a numerical discretization for solving the SWR problems using a forward Euler discretization in time and a finite volume method in space, including a local Newton iteration for solving the nonlinear transmission conditions. Our discrete algorithm is asymptotic preserving, i.e., robust in the vanishing viscosity limit. Finally, we present numerical results that confirm the theoretical findings, in particular the convergence of the algorithm. Moreover, we show that the SWR algorithm can be successfully applied to two-phase flow problems in porous media as paradigms for evolution equations with strongly nonlinear advective and diffusive fluxes.

Flux-mortar mixed finite element methods with multipoint flux approximation

The flux-mortar mixed finite element method was recently developed in Boon et al. (2022) for a general class of domain decomposition saddle point problems on non-matching grids. In this work we develop the method for Darcy flow using the multipoint flux approximation as the subdomain discretization. The subdomain problems involve solving positive definite cell-centered pressure systems. The normal flux on the subdomain interfaces is the mortar coupling variable, which plays the role of a Lagrange multiplier to impose weakly continuity of pressure. We present well-posedness and error analysis based on reformulating the method as a mixed finite element method with a quadrature rule. We develop a non-overlapping domain decomposition algorithm for the solution of the resulting algebraic system that reduces it to an interface problem for the flux-mortar, as well as an efficient interface preconditioner. A series of numerical experiments is presented illustrating the performance of the method on general grids, including applications to flow in complex porous media.

Stochastic multiscale flux basis for Stokes-Darcy flows

Three algorithms are developed for uncertainty quantification in modeling coupled Stokes and Darcy flows. The porous media may consist of multiple regions with different properties. The permeability is modeled as a non-stationary stochastic variable, with its log represented as a sum of local Karhunen-Loève (KL) expansions. The problem is approximated by stochastic collocation on either tensor-product or sparse grids, coupled with a multiscale mortar mixed finite element method for the spatial discretization. A non-overlapping domain decomposition algorithm reduces the global problem to a coarse scale mortar interface problem, which is solved by an iterative solver, for each stochastic realization. In the traditional domain decomposition implementation, each subdomain solves a local Dirichlet or Neumann problem in every interface iteration. To reduce this cost, two additional algorithms based on deterministic or stochastic multiscale flux basis are introduced. The basis consists of the local flux (or velocity trace) responses from each mortar degree of freedom. It is computed for each subdomain independently before the interface iteration begins. The use of the multiscale flux basis avoids the need for subdomain solves on each iteration. The deterministic basis is computed at each stochastic collocation and used only at this realization. The stochastic basis is formed by further looping over all local realizations of a subdomain's KL region before the stochastic collocation begins. It is reused over multiple realizations. Numerical tests are presented to illustrate the performance of the three algorithms, with the stochastic multiscale flux basis showing significant savings in computational cost compared to the other two algorithms.

Explicit/implicit and Crank–Nicolson domain decomposition methods for parabolic partial differential equations

Two parallel non-overlapping domain decomposition algorithms for solving parabolic partial differential equations are proposed. The algorithms combine Crank–Nicolson scheme with implicit Galerkin finite element methods in sub-domains and explicit flux approximation along inner boundaries at each time step. Thus, parallelism can be easily achieved. L2-norm error estimates for these explicit/implicit procedures are presented, in which time step constraints are proved to be less severe than that of fully explicit schemes. Numerical experiments are also performed to verify the theoretical analysis.

Non-overlapping domain decomposition algorithm based on modified transmission conditions for the Helmholtz equation

A square-root based transmission conditions domain decomposition method was recently introduced for the Helmholtz equation. It produces an effective algorithm where the convergence is independent of the wavenumber and the mesh discretization. We modify here these conditions in order to guarantee well-posedness of local problems and further improve the efficiency of the whole method. Numerical results, in particular in the three dimensional case, show significant reduction of the computational time needed in the iterative procedure while preserving the iteration number when compared with the original algorithm.

A multiscale flux basis for mortar mixed discretizations of Stokes–Darcy flows

A multiscale flux basis algorithm is developed for the Stokes–Darcy flow problem. The method is based on a non-overlapping domain decomposition algorithm, where the global problem is reduced to a coarse scale mortar interface problem that is solved by an iterative solver. Subdomain solves are required at each interface iteration, so the cost for the method without a multiscale basis can be high when the number of subdomains or the condition number of the interface problem is large. The proposed algorithm involves precomputing a multiscale flux basis, which consists of the flux (or velocity trace) response from each mortar degree of freedom. It is computed by each subdomain independently before the interface iteration begins. The subdomain solves required at each iteration are substituted by a linear combination of the multiscale basis. This may lead to a significant reduction in computational cost since the number of subdomain solves is fixed, depending only on the number of mortar degrees of freedom associated with a subdomain. Several numerical examples are carried out to demonstrate the efficiency of the multiscale flux basis implementation for large-scale Stokes–Darcy problems.

Block steepest descent method based on nonoverlapping domain decomposition, I

AbstractIn the present paper we propose a new iterative method for the numerical solution of system of linear algebraic equations with symmetric positive definite and positive semidefinite matrices arising from the approximation of diffusion equations by the mixed hybrid finite element method on triangular and tetrahedral meshes. The method is based on a combination of the block steepest descent idea and nonoverlapping domain decomposition algorithm. We give the formulation of the method and a simple convergence estimate in the case of regular shaped quasiuniform meshes.

A Global Jacobian Method for Mortar Discretizations of Nonlinear Porous Media Flows

We describe a nonoverlapping domain decomposition algorithm for nonlinear porous media flows discretized with the multiscale mortar mixed finite element method. There are two main ideas: (1) linearize the global system in both subdomain and interface variables simultaneously to yield a single Newton iteration; and (2) algebraically eliminate subdomain velocities (and optionally, subdomain pressures) to solve linear systems for the 1st (or the 2nd) Schur complements. Solving the 1st Schur complement system gives the multiscale solution without the need to solve an interface iteration. Solving the 2nd Schur complement system gives a linear interface problem for a nonlinear model. The methods are less complex than a previously developed nonlinear mortar algorithm, which requires two nested Newton iterations and a forward difference approximation. Furthermore, efficient linear preconditioners can be applied to speed up the iteration. The methods are implemented in parallel, and a numerical study is performed to compare convergence behavior and parallel efficiency.

Coupling discontinuous Galerkin discretizations using mortar finite elements for advection–diffusion–reaction problems

We investigate DG–DG domain decomposition coupling using mortar finite elements to approximate the solution to general second-order partial differential equations. We weakly impose an inflow boundary condition on the inflow part of the interface and the Dirichlet boundary condition on the elliptic part of the interface via Lagrange multipliers. We obtain the matching condition by imposing the continuity of the total flux through the interface and the continuity of the solution on the elliptic parts of the interface. The diffusion coefficient is allowed to be degenerate and the mortar interface couples efficiently the multiphysics problems. The (discrete) problem is solvable in each subdomain in terms of Lagrange multipliers and the resulting algorithm is easily parallelizable. The subdomain grids need not match and the mortar grid may be much coarser, giving a two-scale method. Convergence results in terms of the fine subdomain scale h and the coarse mortar scale H are then established. A non-overlapping parallelizable domain decomposition algorithm (Arbogast et al., 2007) reduces the coupled system to an interface mortar problem. The properties of the interface operator are discussed.

The fractional steps domain decomposition method for numerical solution of a class of viscous wave equations

In this article, an efficient fractional steps domain decomposition method (FSDDM) is derived for parallel numerical solution of a class of viscous wave equations. In this procedure, the large domain is divided into multiple block sub-domains. The values on the interfaces of sub-domains are found by an efficient local multilevel scheme, implicit scheme is used for computing the interior values in sub-domains. Some techniques, such as non-overlapping domain decomposition, fractional steps and extrapolation algorithm are adopted. Numerical experiments are performed to demonstrate the efficiency and accuracy of the method.

A Nonoverlapping Domain Decomposition Method for Incompressible Stokes Equations with Continuous Pressures

A nonoverlapping domain decomposition algorithm is proposed to solve the linear system arising from mixed finite element approximation of incompressible Stokes equations. A continuous finite element space for the pressure is used. In the proposed algorithm, Lagrange multipliers are used to enforce continuity of the velocity component across the subdomain boundary. The continuity of the pressure component is enforced in the primal form, i.e., neighboring subdomains share the same pressure degrees of freedom on the subdomain interface and no Lagrange multipliers are needed. After eliminating all velocity variables and the independent subdomain interior parts of the pressures, a symmetric positive semidefinite linear system for the subdomain boundary pressures and the Lagrange multipliers is formed and solved by a preconditioned conjugate gradient method. A lumped preconditioner is studied and the condition number bound of the preconditioned operator is proved to be independent of the number of subdomains fo...

A quasi-optimal non-overlapping domain decomposition algorithm for the Helmholtz equation

This paper presents a new non-overlapping domain decomposition method for the Helmholtz equation, whose effective convergence is quasi-optimal. These improved properties result from a combination of an appropriate choice of transmission conditions and a suitable approximation of the Dirichlet to Neumann operator. A convergence theorem of the algorithm is established and numerical results validating the new approach are presented in both two and three dimensions.

The Hermite radial basis function control volume method for multi-zones problems; A non-overlapping domain decomposition algorithm

In this paper a non-overlapping non-iterative multi-domain formulation for the control volume Hermite radial basis functions (CV-HRBF) method is proposed, where the local Hermitian RBF meshless collocation method is used to satisfy a physical matching condition at the sub-domain boundaries. In addition, the robustness of the Hermite interpolation is exploited even further to apply multiple flux continuities for those cases where more than two sub-domains converge in the same point. The algorithm is first validated in one-dimensional advection diffusion problems for which an analytical solution is known. Its accuracy is compared with a classic CV approach and a local radial basis function collocation method (LRBFCM). More general applications in two and three-dimensional domains are then considered. A heat transfer problem in strongly heterogeneous materials, and a groundwater flow problem in presence of geological layers characterised by different hydraulic conductivity, are taken as engineering applications to test the capabilities of the CV-HRBF method to handle multi-zone problems. Finally, the transport of a single species is simulated in a one-dimensional channel consisting of two adjacent zones that feature different Peclet numbers.

Multipliers-free dual-primal domain decomposition methods for nonsymmetric matrices and their numerical testing

The most commonly used nonoverlapping domain decomposition algorithms, such as the FETI-DP and BDDC, require the introduction of discontinuous vector spaces. Most of the works on such methods are based on approaches that originated in Lagrange multipliers formulations. Using a theory of partial differential equations formulated in discontinuous piecewise-defined functions, introduced and developed by Herrera and his collaborators through a long time span, recently the authors have developed an approach to domain decomposition methods in which general problems with prescribed jumps are treated at the discrete level. This yields an elegant and general direct framework that permits analyzing the problems in greater detail. The algorithms derived using it have properties similar to those of well-established methods such as FETI-DP, but, in our experience, they are easier to implement. Also, they yield explicit matrix formulas that unify the different methods. Furthermore, this multipliers-free framework has permitted us to extend such formulas to make them applicable to nonsymmetric matrices. The extension of the unifying matrix formulas to nonsymmetric matrices is the subject matter of the present article. A conspicuous result is that in numerical experiments in 2D and 3D, the MF-DP algorithms for nonsymmetric matrices exhibit an efficiency of the same order as state-of-the-art algorithms for symmetric matrices, such as BDDC, FETI-DP, and MF-DP.© 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1262-1289, 2011

Schwarz waveform relaxation algorithms for semilinear reaction-diffusion equations

We introduce nonoverlapping domain decomposition algorithms of Schwarzwaveform relaxation type for the semilinear reaction-diffusionequation. We define linear Robin and secondorder (or Ventcell) transmission conditions between the subdomains, which we prove tolead to a well defined and converging algorithm. We also propose nonlinear transmission conditions. Both types are basedon best approximation problems for the linear equation and provideefficient algorithms, as the numerical results that we present hereshow.

An efficient domain decomposition method for three-dimensional parabolic problems

A non-overlapping domain decomposition algorithm to solve three-dimensional parabolic partial differential equations is presented. It has been shown in this paper that the algorithm is unconditionally stable and efficient. Spectral radii for the interface and interior region are provided. Unlike two-dimensional problem, it has been found out that estimating the values of the points of the interface in three-dimensional problem is no longer negligible.

Coupling Discontinuous Galerkin and Mixed Finite Element Discretizations using Mortar Finite Elements

Discontinuous Galerkin (DG) and mixed finite element (MFE) methods are two popular methods that possess local mass conservation. In this paper we investigate DG-DG and DG-MFE domain decomposition couplings using mortar finite elements to impose weak continuity of fluxes and pressures on the interface. The subdomain grids need not match, and the mortar grid may be much coarser, giving a two-scale method. Convergence results in terms of the fine subdomain scale $h$ and the coarse mortar scale $H$ are established for both types of couplings. In addition, a nonoverlapping parallel domain decomposition algorithm is developed, which reduces the coupled system to an interface mortar problem. The properties of the interface operator are analyzed.

Numerical modelling of the weight-bearing total knee joint replacement and usage in practice

In the contribution a weight-bearing total knee joint replacement will be based on numerical results on a non-linear contact problem with Coulombian friction in elasticity. The non-overlapping domain decomposition algorithm will be used. The main goal of the contribution represents an application of mathematical modelling and obtained numerical results to the practice—the total knee joint replacement.